The action of the group of gauge transformations of the symplectic frame bundle of \(M\)  on the space of affine connections, possibly having torsion, preserving \(\Omega \) , is Hamiltonian with a moment map constructed in the manner of Atiyah and Bott in [1]}. See Theorem REF  for the precise statement. Using the symplectic form the Lie algebra of infinitesimal gauge transformations can be identified with the space \(\Gamma (S^{2}(T^{\ast }M))\)  equipped with a constant multiple of the algebraic Lie bracket mentioned above. The obstruction to this Lie algebra acting on \(\mathbb {S}(M, \Omega )\)  is given by the Schouten bracket. However, an extended bracket on \(\Gamma (S^{2}(T^{\ast }M)) \oplus \Gamma (S^{3}(T^{\ast }M))\)  built from the Poisson bracket and the algebraic bracket does act on \(\mathbb {S}(M, \Omega )\) , by symplectic affine transformations. This extended action can be combined in a coherent way with the action of \(\operatorname{ham}(M, \Omega )\)  on \(\mathbb {S}(M, \Omega )\) . While the resulting action is only weakly Hamiltonian, the cocycle measuring nonequivariance of the moment map can be computed explicitly and from it there is constructed the Hamiltonian action of \((\widehat{\mathfrak {h}}, (\!(\,\cdot \,, \,\cdot \,)\!)_{s, t})\) . This action combines, at the infinitesimal level, the actions of \(\operatorname{Ham}(M, \Omega )\)  and the group of gauge transformations to yield a Hamiltonian action with a moment map \(\widehat{\mathbb {M}}\)  that simultaneously extends the moment map \({K}\)  for the action of the Lie algebra \(\operatorname{ham}(M, \Omega )\)  of Hamiltonian vector fields and the Atiyah–Bott moment map.
